Best Known (104, 104+19, s)-Nets in Base 5
(104, 104+19, 43404)-Net over F5 — Constructive and digital
Digital (104, 123, 43404)-net over F5, using
- 51 times duplication [i] based on digital (103, 122, 43404)-net over F5, using
- net defined by OOA [i] based on linear OOA(5122, 43404, F5, 19, 19) (dual of [(43404, 19), 824554, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5122, 390637, F5, 19) (dual of [390637, 390515, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5122, 390642, F5, 19) (dual of [390642, 390520, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(5121, 390625, F5, 19) (dual of [390625, 390504, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(5105, 390625, F5, 17) (dual of [390625, 390520, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(51, 17, F5, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(5122, 390642, F5, 19) (dual of [390642, 390520, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5122, 390637, F5, 19) (dual of [390637, 390515, 20]-code), using
- net defined by OOA [i] based on linear OOA(5122, 43404, F5, 19, 19) (dual of [(43404, 19), 824554, 20]-NRT-code), using
(104, 104+19, 195322)-Net over F5 — Digital
Digital (104, 123, 195322)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5123, 195322, F5, 2, 19) (dual of [(195322, 2), 390521, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5123, 390644, F5, 19) (dual of [390644, 390521, 20]-code), using
- construction XX applied to Ce(18) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- linear OA(5121, 390625, F5, 19) (dual of [390625, 390504, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(5105, 390625, F5, 17) (dual of [390625, 390520, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(597, 390625, F5, 16) (dual of [390625, 390528, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(51, 18, F5, 1) (dual of [18, 17, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(18) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(5123, 390644, F5, 19) (dual of [390644, 390521, 20]-code), using
(104, 104+19, large)-Net in Base 5 — Upper bound on s
There is no (104, 123, large)-net in base 5, because
- 17 times m-reduction [i] would yield (104, 106, large)-net in base 5, but